Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Abstraction Principles and the Size of Reality

Bokai Yao

TL;DR

This work analyzes Fregean abstraction principles within set theories that include urelements, revealing how the size of reality (the number of urelements) governs the behavior of Basic Law V and Hume's Principle. It proves that $BLV$ becomes equivalent to the urelements forming a set under certain axioms, while plenitude and other choice-like principles can preclude definable $BLV$ maps in models with many urelements; it also demonstrates independence phenomena for $HP$ for sets, including models where $RP$ holds but $HP$ for sets fails, and constructs KMU-like models with global well-orderings that lack parameter-free $HP$ maps for classes. For classes, the results show $HP$ can be secured under Limitation of Size with a global well-order, yet there exist strong-models (assuming an inaccessible cardinal) where no such parameter-free definable map exists. Overall, the paper highlights how the abundance or scarcity of urelements shapes the applicability and definability of fundamental abstraction principles, offering insights into the interplay between Fregean logic and urelement-based set theory.

Abstract

The Fregean ontology can be naturally interpreted within set theory with urelements, where objects correspond to sets and urelements, and concepts to classes. Consequently, Fregean abstraction principles can be formulated as set-theoretic principles. We investigate how the size of reality-i.e., the number of urelements-interacts with these principles. We show that Basic Law V implies that for some well-ordered cardinal $κ$, there is no set of urelements of size $κ$. Building on recent work by Hamkins \cite{hamkins2022fregean}, we show that, under certain additional axioms, Basic Law V holds if and only if the urelements form a set. We construct models of urelement set theory in which the Reflection Principle holds while Hume's Principle fails for sets. Additionally, assuming the consistency of an inaccessible cardinal, we produce a model of Kelley-Morse class theory with urelements that has a global well-ordering but lacks a definable map satisfying Hume's Principle for classes.

Abstraction Principles and the Size of Reality

TL;DR

This work analyzes Fregean abstraction principles within set theories that include urelements, revealing how the size of reality (the number of urelements) governs the behavior of Basic Law V and Hume's Principle. It proves that becomes equivalent to the urelements forming a set under certain axioms, while plenitude and other choice-like principles can preclude definable maps in models with many urelements; it also demonstrates independence phenomena for for sets, including models where holds but for sets fails, and constructs KMU-like models with global well-orderings that lack parameter-free maps for classes. For classes, the results show can be secured under Limitation of Size with a global well-order, yet there exist strong-models (assuming an inaccessible cardinal) where no such parameter-free definable map exists. Overall, the paper highlights how the abundance or scarcity of urelements shapes the applicability and definability of fundamental abstraction principles, offering insights into the interplay between Fregean logic and urelement-based set theory.

Abstract

The Fregean ontology can be naturally interpreted within set theory with urelements, where objects correspond to sets and urelements, and concepts to classes. Consequently, Fregean abstraction principles can be formulated as set-theoretic principles. We investigate how the size of reality-i.e., the number of urelements-interacts with these principles. We show that Basic Law V implies that for some well-ordered cardinal , there is no set of urelements of size . Building on recent work by Hamkins \cite{hamkins2022fregean}, we show that, under certain additional axioms, Basic Law V holds if and only if the urelements form a set. We construct models of urelement set theory in which the Reflection Principle holds while Hume's Principle fails for sets. Additionally, assuming the consistency of an inaccessible cardinal, we produce a model of Kelley-Morse class theory with urelements that has a global well-ordering but lacks a definable map satisfying Hume's Principle for classes.

Paper Structure

This paper contains 7 sections, 14 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Urelement Set Theory
  3. Axioms in urelement set theory
  4. Basic facts in $\textup{ZFU}_\text{R}$
  5. Basic law V in Urelement Set Theory
  6. Hume's Principle in Urelement Set Theory
  7. Hume's Principle in Urelement Class Theory

Key Result

Theorem 1

Let $U$ be a model of $\textup{ZFU}_\text{R}$ + "$\mathscr{A}$ is a set" equipped with definable classes. Then $U$ has a definable map that fulfills Basic Law V.

Theorems & Definitions (30)

  • Theorem 1: hamkins2022fregean
  • proof
  • Lemma 2
  • proof
  • Claim 2.1
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 20 more